Finite Group Databasebeta

← #379642

Subgroups — #379642

Order 512.

954 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23638C211yesmaximal, characteristic
2#20386C212yesmaximal
3#11201C212yesmaximal
4#11097C211yesmaximal, characteristic
5#10910C212yesmaximal, complemented
6#10909C212yesmaximal, complemented
7#10809C212yesmaximal, complemented
8#10809C212yesmaximal, complemented
9#10715C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C2×#427C2211yescharacteristic
13C2×#407C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C2×#405C2212yes—
16C2×#405C2212yes—
17#2280C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19#2077C412yescomplemented
20#2077C412yescomplemented
21#2076C411yescharacteristic, complemented
22#2076C411yescharacteristic, complemented
23#2075C411yescharacteristic, complemented
24#2075C411yescharacteristic, complemented
25#1753—22no—
26#1753—22no—
27#1753—22no—
28#1753—22no—
29C2×#341C2212yes—
30C2×#341C2211yescharacteristic
31#1735C2211yescharacteristic
32#1735C2212yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35C2×#339C2212yes—
36C2×#339C2212yes—
37#1729C2212yes—
38#1729C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C24×C4C2311yescharacteristic
42C22×M16—24no—
43C22×M16—24no—
44C22×M16—41no—
45C22×M16—41no—
46C22×M16D411yescharacteristic
47C22×M16D412yes—
48C22×M16D411yescharacteristic
49C22×M16C2312yes—
50C22×M16C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×#117C2312yes—
55C2×#117C2312yes—
56C22×C22⋊C4—21no—
57C22×C42C2312yes—
58C2×#108—24no—
59C2×#108—41no—
60#427—41no—
61#427—41no—
62C2×#107C2×C411yescharacteristic
63C2×#107C2×C411yescharacteristic
64C2×#106C2×C412yes—
65#407—24no—
66#407—24no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—41no—
70C2×C22⋊C8—41no—
71C2×C22⋊C8C2×C411yescharacteristic
72C2×C22⋊C8C2×C411yescharacteristic
73C2×C22⋊C8C2×C411yescharacteristic
74C2×C22⋊C8C2×C411yescharacteristic
75#405—24no—
76#405—24no—
77#405—42no—
78#405—42no—
79C2×#97—22no—
80C2×#97—22no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83#381—22no—
84#380C2×C412yes—
85#380C2×C412yes—
86#380C2×C412yes—
87#380C2×C412yes—
88#379—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—41no—
92C4×C22⋊C4—41no—
93C2×#95—21no—
94#341—24no—
95#341—41no—
96#340C2×C411yescharacteristic
97#340C2×C411yescharacteristic
98#340C2×C412yes—
99#340C2×C412yes—
100#340C2×C412yes—
101#340C2×C412yes—
102#340C2×C412yes—
103#339—24no—
104#339—24no—
105#339—42no—
106#339—42no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—24no—
112#336—24no—
113C25#4111yescharacteristic
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4#4111yescharacteristic
128C23×C4C2411yescharacteristic
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—41no—
144C2×M16—42no—
145C2×M16—41no—
146C2×M16—44no—
147C2×M16—48no—
148C2×M16—44no—
149C2×M16#4114yes—
150C2×M16#4114yes—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C22×C411yescharacteristic
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C411yescharacteristic
168C22×C8C22×C411yescharacteristic
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8#4114yes—
172C22×C8#4114yes—
173#117—22no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—42no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201#108—44no—
202#107—41no—
203#107—41no—
204#106—42no—
205C22⋊C8—28no—
206C22⋊C8—28no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211#97—44no—
212#97—44no—
213C4×C8—44no—
214C4×C8—44no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—42no—
220C24—21no—
221C24—21no—
222C24—22no—
223C24—22no—
224C24—42no—
225C24—41no—
226C24—41no—
227C24C4×D411yescharacteristic
228C24C4×D411yescharacteristic
229C24C2×#4111yescharacteristic
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C4×D411yescharacteristic
272C22×C4C4×D411yescharacteristic
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4C2×#4112yes—
277C22×C4C2×#4112yes—
278C22×C4C2×#4112yes—
279C22×C4H(5,𝔽2)11yescharacteristic
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C42−1+411yescharacteristic
283M16—24no—
284M16—24no—
285M16—24no—
286M16—24no—
287M16—44no—
288M16—48no—
289M16—44no—
290M16—42no—
291M16—42no—
292M16—42no—
293M16—42no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—41no—
322C2×C8#12314yes—
323C2×C8#12314yes—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C22⋊C4—48no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C23—22no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—22no—
355C23—22no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—22no—
361C23—22no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23—42no—
368C23—41no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23C2×C4×D411yescharacteristic
374C23#52912yes—
375C23#52912yes—
376C23#53811yescharacteristic
377C23#53911yescharacteristic
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4C4×#4112yes—
417C2×C4C4×#4112yes—
418C2×C4#51811yescharacteristic
419C2×C4#52011yescharacteristic
420C2×C4C2×#12312yes—
421C2×C4#52512yes—
422C2×C4#53411yescharacteristic
423C2×C4#53711yescharacteristic
424C2×C4#55111yescharacteristic
425C2×C4#55111yescharacteristic
426C2×C4#55112yes—
427C2×C4#55111yescharacteristic
428C2×C4#55212yes—
429C2×C4#55312yes—
430C2×C4#55312yes—
431C2×C4#55512yes—
432C2×C4#55612yes—
433C2×C4#55911yescharacteristic
434C8—24no—
435C8—24no—
436C8—44no—
437C8—41no—
438C8—41no—
439C8—41no—
440C8—41no—
441C8—44no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—84no—
461C22—82no—
462C22—82no—
463C22C4×#12312yes—
464C22#232511yescharacteristic
465C22#251711yescharacteristic
466C22#251812yes—
467C22#252311yescharacteristic
468C22#295612yes—
469C22#295711yescharacteristic
470C22#295711yescharacteristic
471C4—41no—
472C4—41no—
473C4—84no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4—82no—
478C4#237211yescharacteristic
479C4#237312yes—
480C4#237411yescharacteristic
481C4#248311yescharacteristic
482C4#248512yes—
483C4#260011yescharacteristic
484C2—22no—
485C2—21no—
486C2—21no—
487C2—81no—
488C2—81no—
489C2#1424811yescharacteristic
490C2#2159412yes—
491C2#2159412yes—
492C2#2220611yescharacteristic
493C2#2220711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×M16 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #10910C2semidirect
6. #10909C2semidirect
7. #10809C2semidirect
8. #10809C2semidirect
19. #2077C4semidirect
20. #2077C4semidirect
21. #2076C4semidirect
22. #2076C4semidirect
23. #2075C4semidirect
24. #2075C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#236382
#203862
#112012
#110972
#109102
#109092
#108092
#107152